There are three distinct types of spaces in the decomposition (4.1). We will refer to these different types of subspaces, suggestively, as: gauge-like spaces, F-spaces, and H-spaces. Naturally, the gauge-like spaces will be the complexified Lie algebras themselves as, like gauge fields, they transform in the adjoint representation. The F-spaces transform either as a singlet or in the (anti-)fundamental representation of our unitary groups, hence the suggestive “F” to denote either the “fermion-like” gauge transformations. Finally the H-spaces are quite different from the rest of the spaces so far considered. They are real vector spaces in terms of scalar multiplication, yet still transform in the complex fundamental representation of SU(2)×SU(3). We show in subsection 4.5.3 that invariants formed with these irreducible representation spaces can be understood in terms of six dimensional complex vector spaces transforming as triplets of SU(3) and doublets of SU(2).
4.5 Comments on Decomposition 55
We will show that due to this feature, the H−spaces mirror certain properties of the Standard Model Higgs, but under different gauge groups. Consequently, we have used the letter “H” to denote that these representation spaces, to the extent that we can make comparisons between objects transforming under different groups, are “Higgs-like”.
Decomposition (4.1) only has four parameters in its definition. These are the param-eters α, β 6= ±1, and the two parameter describing the relative charge assignment of our U(1) field. The irreducible representation spaces contained in the decomposition remain the same under any choice of these parameters, but this is not the case for invariants formed of different irreducible representation spaces. Indeed, changes to these parameters will be equivalent to scalar multiplication of these invariants. Thus the parametersαandβ suggestively encode interaction strengths between irreducible representation spaces. With-out a full theory it is of course impossible to make precise statements abWith-out interactions;
however, we will show in subsection 4.5.3 that the specific realization of the representations does indicate distinct coupling values.
4.5.1 Gauge-like Subspaces
For adjoint representations we have the complexification of the Lie algebras u(1), su(2), and su(3). The su(N) Lie algebras arise from restricted representations of the automor-phism groupG2, and their complexification is required to fully span the algebra M(C⊗O).
The U(1) transformations do not originate from the automorphism group of the Octonions.
Instead, this redundancy corresponds to the fact that invariants constructed out of the ir-reducible representation spaces of SU(2)×SU(3) in M(C⊗O) are invariant under unitary transformation, not just special unitary transformations. This, combined with the require-ments of linear independence and a full direct sum decomposition of M(C⊗O), demands the existence of a U(1) transformation acting on the same decomposition as SU(3) and a corresponding C⊗u(1) subspace of M(C⊗O). Resultantly, it is valid to say that, within the setup shown here, the appearance of the SU(2)×SU(3) symmetry structure demands the existence of an additional U(1) symmetry.
One important point of discussion is that gauge fields in the Standard Model take values in the real Lie algebras, and not their complexification. However, as stated before, the corresponding representations are still those of the (special)unitary groups. This implies that, for example, the linearly independent subspacessu(3) andisu(3) are both irreducible representation spaces of the group SU(3). It is clear that for agreement with Standard Model representations there must then be some selection mechanism by which one of these irreducible representation spaces does not appear as associated to a physical field. However finding such a mechanism would require both the merger of our gauge representations with Lorentz representations, as gauge fields are described by objects which are both spacetime and Lie algebra vectors, and the formulation of a theory in which such a selection mechanism can be defined. As both the formulation of a full theory and simultaneous incorporation of gauge and Lorentz representations are beyond the scope of this paper, so is the discussion of the complexification of our Lie algebras. We do note that the appearance of the complexified Lie algebras is similar to the appearance of gauge fields in
NCG, and we comment on this in subsection 5.1.3.
4.5.2 The F - Spaces
To offer a concise presentation of the irreducible representation spaces contained in the decomposition (4.1), we write out the set of fundamental representation spaces contained in the F - subspaces of M(C⊗O) as
F3 contains 4×(3,1) (4.20)
F3∗ contains 4×(3∗,1) (4.21)
F1 contains (1,2) + (1,2∗) + 4×(1,1) (4.22) F1∗ contains (1,2∗) + (1,2) + 4×(1,1) (4.23) with the notation (a, b) describing a space transforming in thearepresentation of SU(3) and b representation of SU(2). As before we have omitted any U(1) charge assignment, as the relative charges are not fixed. Note that in (4.20)-(4.23) we have no particles simultaneously transforming under SU(3) and SU(2). Thus, it is clear that in this construction not all representation spaces are realized in the direct sum decomposition of M(C⊗O). Further still the fundamental representations of (4.20)-(4.23) appear in distinct multiplicities.
These distinct multiplicities of the irreducible representation spaces in the F−spaces arise as a direct consequence of simultaneously incorporating both adjoint and fundamental representations within the same algebra of linear maps. This can be seen from the ori-gin of our F−spaces as the irreducible representation spaces which were trivially linearly independent from either su(2) or su(3) by virtue of belonging to different ideals. These two Lie algebras themselves belong to ideals of distinct dimensions, i.e. are represented by matrices of different size. Explicitly, in the SU(2) decomposition we had 4 singlets, while in the SU(3) decomposition we only had two singlets. Therefore it is no surprise that we obtain different multiplicities of the different fundamental representation spaces.
In the Standard Model not all fundamental representations of our gauge groups are present. There are no right handed doublets of SU(2). This implies that while we observe the representation (3,2), and the corresponding conjugate representation, we do not observe any particles transforming in the (3∗,2), or the conjugate, representation. For comparison with Standard Model representations, it is an attractive feature that not all fundamental representations appear in the direct sum decomposition. Of course, in our set up we do not have either of the representations (3,2), (3,2∗), or their conjugates. This, in addition to the lack of Lorentz representations, means we cannot draw a direct comparison with our representations and those of the Standard Model. However, we still find the general feature of different multiplicities of representation spaces. We highlight this natural appearance of distinct multiplicities of gauge groups as a feature of interest in comparing sets of representation spaces with those of the Standard Model.10
10Note here that having distinct multiplicities also considers representation spaces which do not appear at all in the direct sum decomposition, as these representation spaces have multiplicity zero.
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4.5.3 H - Spaces
The remaining spaces to analyse in our decomposition (4.1) are H1α and H2β. Due to the similarity in the structure of the spaces H1α and H2β, we will only analyse the space H1α, with similar arguments applicable for H2α. While the complex structure in H1α is not simply multiplication byi, we can understand the space in terms of mapsφ: 3→2, where indeed the complex structure is multiplication byi, see subsection 4.4.3. According to the definition of H1α in (4.15), elements of this space are maps h on C⊗O which map 3 to 2 and 3∗ to 2∗ such that
h=φ+αφ∗, (4.24)
where φ∗ : 3 →2. As φ and φ∗ belong to different left and right ideals of M(C⊗O), any invariants formed with h and irreducible representation spaces of Fi reduce to invariants formed with these irreducible representations in Fi and eitherφ or φ∗. Then as all the 12 real parameters ofH1αare encoded in the 6 complex parameters ofφ, one can fully describe the different possible map compositions with H1α by the set of complex parameters in φ.
Now asα 6=±1, there are no maps inH1αwhose conjugate map is alsoH1α. Further there is no independent conjugate space toH1α in the direct sum decomposition (4.1).11 However, we still have that for the mapφ, which transforms in the fundamental representation, there is always a conjugate map φ∗ transforming in the complex conjugate representation.
In the case one were to write down a theory of interacting fields transforming in irre-ducible representations, one could then parametrize all the interactions and dynamics of H1α in terms of the mapsφ transforming in the fundamental representation. This would of course require also including φ∗, but without independent dynamics and not as a separate field. In other words, φ∗ would only be required to accurately describe interactions be-tween fields. In the Standard Model, Yukawa interactions require the explicit appearance of both the Higgs doublet Φ and the conjugate Φ∗, which has opposite charge assignments to Φ, Ref. [54]. Thus by viewing our spaceH1α as described by mapsφ transforming in the fundamental representations of our gauge groups, we recover a picture which, to the extent that we can draw comparisons to field theory, is quite similar to the Standard Model Higgs doublet. Of course in the Standard Model the Higgs doublet and its conjugate are needed to give masses to the different components of the left-handed SU(2) doublets, for example to up- and down-type quarks. This is not possible in our construction asφ andφ∗ not only live in different ideals but also act on the non-similar 3 and 3∗ representations of SU(3).12 Instead the respective mapsφandφ∗must form contractions between respective conjugate representations.
11Note that even forα=i, under complex conjugationH1i→H1−i=iH1i. The spaceH1αis a real vector space, and it is simple to check that it isR-linearly independent from the space iH1α for anyα∈C.
12The Higgs doublet transforms under SU(2) as its only non-abelian gauge group. Since the conjugate representation 2∗is related by a similarity transformation to the doublet representation 2, this means that the Higgs doublet and its conjugate may still form invariants with fields transforming in the ¯2 of SU(2).
Both of these invariants are required, as after spontaneous symmetry breaking the Higgs doublet gives mass to one component of the fermionic SU(2) doublet while the conjugate Higgs doublet gives mass to the other component of the fermionic SU(2) doublet.
When considering such contractions the factor α may then be interpreted as a relative coupling parameter between φ and φ∗. That is, as a coupling that incorporates a dis-crimination between the fundamental and anti-fundamental representations. We highlight this discrimination between conjugate representations as a feature of interest, as nature certainly seems to discriminates between these representations. Indeed, our universe is composed primarily of matter and not anti-matter. In the Standard Model, the Higgs dou-blet also couples to up- and down-type quarks with different coupling strengths. Therefore it would be interesting to see whether, in another set-up, one may recover similar spaces to H1αwhich only transform under the non-abelian group SU(2). In such a case the parameter α could potentially become interpreted as a relative coupling strength between elements with opposite weak charge. This would present a natural mechanism by which to generate different masses of particles.
In the above analysis we discussed features of the H-spaces in comparison to the Stan-dard Model Higgs doublet. Naturally there are limitations to how close a connection we can make. One feature which is essential to the Higgs doublet is its scalar nature under Lorentz transformations. This is not a feature on which we can provide comparison as our direct sum decomposition contains no Lorentz representations. Further, any discussion about dynamics is also not possible, since we have no theory for our sets of representations. This prevents any discussion regarding spontaneous symmetry breaking. As such, the above arguments are focused only on the statements we can make about representations. These are: for the map φ complex conjugation results in a representation space with opposite charges, just as is the case for the Higgs doublet; the maps φ and φ∗ are described by the same real components, just like the Higgs doublet and its complex conjugate; and finally φ and φ∗ form invariants with different representations, this is also the case for the Higgs doublet.13 Thus for the available points of comparison there is a strong similarity between the H-spaces and Standard Model Higgs doublet. To make or more definitive statements requires inclusion of spacetime symmetries and the formulation of an action.
We conclude our analysis of theseH-spaces by noting that their appearance is uniquely required by the appearance of our adjoint representations. This arises as a consequence of the H-spaces belonging to the same left ideals as C⊗su(3) and the same right ideals as C⊗su(2). This differs from our F-spaces which share an ideal with at most one of the non-abelian adjoint representations. This suggests the appearance of such H-spaces is not unique to the setup shown here, and could correspond to a more general feature of incorporating irreducible representation spaces as linearly independent maps between restricted representations.
4.5.4 Uniqueness of Decomposition
In this subsection we discuss uniqueness up to linear combination of maps. That is to say, instead of considering F3 and F3∗ we could consider linear combinations of these
13As right handed up- and down-type quarks are singlets of SU(2) with different U(1) charges they are distinct irreducible representation spaces of the Standard Model gauge group.